A working example wrt the supremum norm

[bugatti79]

Folks,

Could anyone give me a working example of a sequence of functions that converges to a function wrt to the supremum norm?

Thank you.

 

[micromass]

Depends on what you mean with "working example". A nice example is probably

[tex]f_n(x)=\sum_{k=0}^n{\frac{x^k}{k!}}.[/tex]

This converges to [itex]f(x)=e^x[/itex]. This convergence is uniform on each bounded set.

 

[bugatti79]

Depends on what you mean with "working example". A nice example is probably

[tex]f_n(x)=\sum_{k=0}^n{\frac{x^k}{k!}}.[/tex]

This converges to [itex]f(x)=e^x[/itex]. This convergence is uniform on each bounded set.

Would this converge to the same function wrt to the integral norm?

 

[micromass]

Yes. It is actually a nice exercise to show that it does. In fact, it suffices in this case to show that

[tex]\int_a^b|f_n|\rightarrow \int_a^b |f|[/tex]

 

[LCKurtz]

Pick a number a with ##0 < a < 1##. Let ##f_n(x) = x^n\hbox{ on }[0,a]##. Then ##\|f_n - 0\|\rightarrow 0##. Note that this fails if ##a=1##.

 

[micromass]

LCKurtz' example is a really nice one! It is nice to notice that it can be generalized into what is known as Dini's theorem:

If [itex](X,d)[/itex] is a compact metric space and if [itex]f_n:X\rightarrow \mathbb{R}[/itex] are a sequence of functions such that
- They are monotonically decreasing/increasing
- They pointswize converge to a function f
- f is continuous
then the convergence is uniform.

It is one of the few cases where pointsiwize convergence implies uniform convergence.

LCKurtz' example follows with X=[0,a] (with 0<a<1), [itex]f_n(x)=x^n[/itex] and [itex]f(x)=0[/itex]. It is a nice exercise to see what goes wrong in the theorem if a=1.

 

[bugatti79]

Depends on what you mean with "working example". A nice example is probably

[tex]f_n(x)=\sum_{k=0}^n{\frac{x^k}{k!}}.[/tex]

This converges to [itex]f(x)=e^x[/itex]. This convergence is uniform on each bounded set.

Would this converge to the same function wrt to the integral norm?

Would it be something like this...


[itex]\displaystyle ||f_n(x)||_1=\int_{a}^{b}| \sum_{n=0}^{\infty} \frac{x^k}{k!}|dx[/itex] for c[a,b]

 

[bugatti79]

Pick a number a with ##0 < a < 1##. Let ##f_n(x) = x^n\hbox{ on }[0,a]##. Then ##\|f_n - 0\|\rightarrow 0##. Note that this fails if ##a=1##.

Hmmm..but regarding post 3 ie a function f_n the converges to the same function f on both sup and integral norms...this function does not converge to a function but to zero...?

LCKurtz' example is a really nice one! It is nice to notice that it can be generalized into what is known as Dini's theorem:

If [itex](X,d)[/itex] is a compact metric space and if [itex]f_n:X\rightarrow \mathbb{R}[/itex] are a sequence of functions such that
- They are monotonically decreasing/increasing
- They pointswize converge to a function f
- f is continuous
then the convergence is uniform.

It is one of the few cases where pointsiwize convergence implies uniform convergence.

LCKurtz' example follows with X=[0,a] (with 0<a<1), [itex]f_n(x)=x^n[/itex] and [itex]f(x)=0[/itex]. It is a nice exercise to see what goes wrong in the theorem if a=1.